Less than 3 percent of illegal immigrants will ever be deported, and more than 700 miles of the U.S.-Mexico border remained unsecured as of 2014, according to Sen. Tom Coburn’s final oversight report released Saturday morning, which found the Homeland Security Department failing in several of its top missions.

The report also said corruption is a serious problem in the Border Patrol, but said agency officials actually told internal affairs investigators to cut down on the number of cases they were pursuing, according to the former division head.

In another finding Mr. Coburn’s staff on the Senate Homeland Security Committee found mission creep to be a problem: agents at one immigration agency spent time cracking down on women’s lingerie that they believed infringed on Major League Baseball’s officially licensed logos. The agents raided a lingerie store in Kansas City, Mo., flashed their badges and confiscated 18 pairs of underwear marked with an unauthorized Kansas City Royals logo, Mr. Coburn’s investigators found.

Mr. Coburn said that agency, U.S. Immigration and Customs Enforcement, should spend more time focusing on illegal immigrants and less time on property issues like women’s underwear.

Meanwhile, more than 700 miles of the border were deemed porous because there was “little to no deployment density or aviation surveillance coverage” to detect illegal immigrants, smugglers or others, said Mr. Coburn, an Oklahoma Republican who retired effective Saturday.

That 700-mile figure accounts for more than a third of the southern border. Mr. Coburn warned the northern border was even worse-off.

“With these broad gaps in coverage of both our Southern and Northern borders, the problem of people and goods illegally entering our country remains a significant concern, and a committed adversary seeking illegal entry into the United States has a reasonable chance of doing so undetected,” said the senator, who was the top Republican on the Homeland Security Committee.

Overall Mr. Coburn, who has studied the department possibly more closely than any other member of Congress, said it hides important data from Congress and the public, and fails to follow cybersecurity protocols itself, even though it is the agency charged with overseeing the issue nationally.

He said the student visa programs have proved to be a terrorist loophole, with about three dozen convicted terrorists using that method for gaining entry into the U.S., according to Homeland Security data. Each ICE agent assigned to the student visa program has to monitor about 6,500 students, which makes the program tough to patrol, he said.

Homeland Security Department spokeswoman Marsha Catron thanked Mr. Coburn for his report, but said it didn’t capture the extent of the work her department does.

“Dr. Coburn’s report on DHS overlooks much of the concrete and recent progress we have made over the past year to improve homeland security and the manner in which DHS conducts business,” she said.

She provided a fact sheet that claimed the border “is safer and more secure than at any time in our nation’s history,” pointing to the record number of agents patrolling between ports of entry. The fact sheet also claimed illegal entries, as measured by number of persons caught at the border, is down — overlooking a recent rise that even top officials such as Secretary Jeh Johnson said caught them off guard.

The fact sheet also points to President Obama’s new deportation amnesty, announced in November, as a sign that the department is getting a handle on illegal immigration. Under that amnesty, up to 5 million illegal immigrants will be granted tentative legal status and work permits to compete legally for job, and most of the rest of the illegal immigrant population will be freed from fear of deportation — though they won’t get official work permits.

The fact sheet also promised more improvements in monitoring student visas, saying that ICE “is developing” new tools that will allow it to punish schools that fail to keep track of their foreign students.